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Quadratic torsion orders on Jacobian varieties

Hamide Kuru, Mohammad Sadek

Abstract

We establish the existence of hyperelliptic curves of genus $g\ge 2$ defined over $\mathbb{Q}$ whose Jacobians possess rational torsion points of order $N$ where $N=4g^2+2g-2$ or $4g^2+ 2g -4$. For $N=2g^2+7g+1$, we introduce a $1$-parameter family of hyperelliptic curves of genus $g$ over $\mathbb{Q}$ with a rational torsion point of order $N$ on their Jacobians.

Quadratic torsion orders on Jacobian varieties

Abstract

We establish the existence of hyperelliptic curves of genus defined over whose Jacobians possess rational torsion points of order where or . For , we introduce a -parameter family of hyperelliptic curves of genus over with a rational torsion point of order on their Jacobians.

Paper Structure

This paper contains 4 sections, 5 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. The construction
  3. The order of the torsion divisor
  4. An infinite family of hyperelliptic curves

Key Result

Proposition 2.1

Fix two integers $g\ge 1$ and $d$, $0\le d\le g-1$. Let $C$ be a hyperelliptic curve defined by the equation $y^2=A(x)^2-\lambda x^{g+1+d}(x-1)^{g-d}$, where $A(x)\in K[x]$, $\deg A(x)\le g$, $\lambda\in K\setminus\{0\}.$ Let $m$ be an integer such that $1\le m < d+1$. Then there is a torsion diviso

Theorems & Definitions (8)

  • Proposition 2.1
  • Proposition 3.1
  • Theorem 3.2
  • Example 3.3
  • Theorem 4.1
  • Example 4.2
  • Corollary 4.3
  • Example 4.4